Part (i) follows from [1]}; here, note that the genus of \(O_k\)  is at least \(2^k-1\) .
Part (ii) follows from results in [2]} and [2]}. We remark that the map in [2]} is defined in terms boundary connected sum, although it applies just the same to the gluing construction detailed above. Again, it is important the number of \({\mathbb {S}}^2\times {\mathbb {S}}^1\)  summands is at least \(2^k-1\)  in the calculation of the stable range.
